It's Not Adding Up

School reforms aren't addressing the real culprit: the American way of teaching.

Mr. Jones's eighth-grade math class is working on geometry. Last Friday the students measured all six interior angles of a hexagon and added up the results. Everyone got a total of approximately 720 degrees. Today they soon agree that that number would stay the same even if the hexagon's shape changed.

Now what? Jones--not his real name--follows up in a typically American way. "There is a formula, and we are going to go through this after spring break. But I will give you a hint right now. If I take the number of sides, and I subtract two, and I multiply that number times 180 degrees, that will tell me how many degrees these add up to. How many sides in this figure?" He pauses. "Six, right? Number of sides, subtract two--gives me what?

Students: "Four."

Jones: "Four. What is four times 180 degrees?"

Jacquille: "Seven hundred twenty."

Jones: "Should be 720, right? How many degrees should there be in a five-sided figure?" He pauses. "Take the formula. The number of sides is five . . . subtract two, and multiply by 180 degrees.

This lesson was videotaped during the 1994-'95 school year, but I'll bet it sounds like your school days, no matter how old you are. We know what went on in Jones's class that day because it had been randomly selected to be part of the Third International Mathematics and Science Study, one of more than 200 eighth-grade math lessons videotaped in Japan, Germany, and the United States.

Jones's lesson wouldn't seem familiar in Japan or Germany. In those countries the class would typically have proceeded in a different direction, according to James Stigler and James Hiebert in The Teaching Gap, which is based in part on the TIMSS videos. The typical Japanese or German teacher wouldn't have just told the students the formula. Instead, he or she would have invited the students to think about various polygons and either led or encouraged them to devise a way of figuring out what their angles add up to. The students might struggle and flounder for a while, but the teacher would have planned and orchestrated the lesson so that in the end they would not only learn the formula but get a feeling for where it comes from and why it works.

Unfortunately American students and teachers rarely explore math concepts in this way. Instead they do cookbook math, applying recipes handed down by the teacher--they memorize a seemingly arbitrary rule and follow it. According to the TIMSS "Synthetic Report" (nces.ed.gov/timss/97198-6.html), "Students were assigned to invent new solutions, proofs, or procedures on their own which require them to think and reason in 44 percent of Japanese [lessons], 4 percent of German lessons, and less than 1 percent of U.S. lessons."

How come? Why do American schools still look, sound, smell, and feel so much like they did when you were a kid? More than 40 years after the Sputnik panic and almost 20 years since the "Nation at Risk" scare, why don't our students learn more? To take one notorious item from the most recent National Assessment of Educational Progress, why can't three out of five American eighth-graders pick, from five options, the correct amount for a 15-percent tip on a restaurant bill?

The usual answers won't do, even though they're all partly true. It's not that teachers are dumb or lazy or that they're underpaid and disrespected. It's not that students are undisciplined or oppressed by silly rules. It's not that classes are too large or too small. It's not that schools are too centralized or too decentralized. It's not that public schools lack competition or that their critics want to destroy them. It's not that the textbooks are too indulgent or too drill happy. It's not even that the standards are too low or too high.

School reforms based on these complaints have kept politicians and lobbyists busy for decades, but they rarely reach the heart of the matter: the American way of teaching isn't doing the job most Americans want done.

The Third International Mathematics and Science Study is the biggest and most sophisticated research project yet in a series of international comparisons that have almost always made American students look bad. In mathematics, eighth-graders from Singapore, Korea, Japan, Canada, France, Australia, Hungary, and Ireland all scored significantly higher than ours (German kids scored about the same).

But this time around, breast-beating wasn't the point. Stigler, who teaches psychology at the University of California at Los Angeles, and Hiebert, who teaches education at the University of Delaware, already knew we were behind. They wanted to learn why, and the videotaped lessons helped them do so. The tapes make it easier than ever before to see how teachers actually teach, rather than guess at what happens in the classroom from recollections, notes, or surveys. Researchers can compare teaching systematically across cultures. And they've found that Japanese, German, and American teachers tend to follow national styles of teaching that differ greatly from one another. Think of how different your best and worst teachers were. Well, those differences pale into insignificance compared to the gap between all of them and their foreign counterparts.

In Germany teachers typically lead students through mathematical concepts, developing procedures for solving different kinds of problems. In Japan teachers appear to take a less active role, but when they give demanding problems to their eighth-graders, they have designed the lesson so that students are likely to use recently learned procedures and to be able to learn from and eventually succeed in their struggles. In the U.S. most teachers don't even aim to teach students to understand math concepts; they aim to teach them how to do things. Hence they proceed as Mr. Jones did: they "present definitions of terms and demonstrate procedures for solving specific problems. Students are then asked to memorize the definitions and practice the procedures." In the TIMSS videotapes, American students spent 96 percent of their desk-work time on routine procedures, compared to 41 percent in Japan.

Questionnaires filled out by teachers confirm the tale of the tapes. According to the TIMSS "Synthetic Report," "Mathematical thinking, such as exploring, developing, and understanding concepts, or discovering multiple solutions to the same problems, was described as the goal of the lesson by 71 percent of the Japanese teachers, compared with 29 percent of German and 24 percent of U.S. teachers."

These expectations shape every aspect of classroom life--even whether teachers use blackboards or overhead projectors. Japanese teachers typically use blackboards and may write over all the boards in the room in the course of a class period, leaving behind an intricately connected trail of thoughts that students can refer to. American teachers favor overhead projectors, which focus students' attention in one place so that they'll pay attention to each step in turn and be able to reproduce all of the steps in practice. Drawing connections is seen as less important. Japanese lessons are as carefully scripted as a theatrical performance; each part refers to earlier parts and the whole makes a complete "story." Japanese teachers explicitly drew connections between different parts in 96 percent of the taped lessons; American and German teachers did so in just 40 percent.

Even seemingly trivial events in the school day are part of this teaching pattern. Stigler and Hiebert and overseas colleagues gathered one day to watch the tape of a U.S. lesson. "The teacher in the video was standing at the chalkboard in the midst of demonstrating a procedure, when a voice came over the public-address system: 'May I have your attention, please. All students riding in bus thirty-one, you will meet your bus in the rear of the school today, not in the front of the school. Teachers please take note of this and remind your students.' A Japanese member of our team reached over and pushed STOP on the VCR. 'What was that?' he asked. 'Oh, nothing,' we replied as we pushed the PLAY button. 'Wait,' protested our Japanese colleague. 'What do you mean, nothing?' As we patiently tried to explain that it was just a P.A. announcement, he became more and more incredulous. Were we implying that it was normal to interrupt a lesson? How could that ever happen? Such interruptions would never happen in Japan, he said, because they would ruin the flow of the lesson." The tapes bore him out. Thirty-one percent of American lessons were somehow interrupted from outside the classroom, but not one Japanese lesson was.

In The Teaching Gap, Stigler and Hiebert stick to what they know--eighth-grade math. Without actually saying so, they imply that some similar indictment applies to American teaching in other subjects and grades as well. They're probably right. For instance, in 1979 University of Illinois researcher Dolores Durkin published a systematic study of 11,587 minutes of American reading classes in third through sixth grades. Just 45 of those minutes were spent on reading-comprehension instruction. Ten times as many minutes were spent testing reading comprehension as on teaching it! What else was going on in those classes besides a dab of reading-comprehension instruction and reading-comprehension testing? Durkin wrote: "Instead of being instructors, the 39 observed teachers were mentioners, assignment givers, assignment duckers, and interrogators."

Teachers' manuals in those grades, Durkin later found, followed a pattern uncannily similar to the one Stigler and Hiebert identify in eighth-grade math. Reading was reduced to a routine of definitions and procedures. The manuals have "a tendency to equate definitions with comprehension instruction and, by so doing, to stop just short of being helpful for reading," she wrote in the 1985 collection Reading Education: Foundations for a Literate America. Students were taught definitions, like the difference between first-person and third-person narration, or the difference between statements of fact and statements of opinion. Then they were to practice distinguishing them on many a worksheet. But the manuals offered teachers no help in showing students how making these distinctions could help them understand stories, ads, or editorials that they read in real life. "It is as if doing little exercises and getting right answers are all that count."

OK, so there's an American way of teaching, and it may not be the greatest. Time for educational experts to issue another manifesto on good teaching practices, right? Wrong, say Stigler and Hiebert. In real life high-minded reform proclamations are usually ignored--and when heeded, they can even make classroom teaching worse. Stigler and Hiebert describe a videotape of an eighth-grade class working its way through a simple problem. At one point the students need to subtract four from one. "Take out your calculators," the teacher says. "Now, follow along with me. Push the one. Push the minus sign. Push the four. Now push the equals sign. What do you get?"

What just happened here? According to the teacher involved, it's an example of educational reform. Not so, insist Stigler and Hiebert--the calculator was just a diversion that "accomplished little on behalf of students' mathematical understanding." They're both right. The National Council of Teachers of Mathematics (NCTM) does recommend that students start using calculators early in their school careers. But as Stigler and Hiebert point out, the reason the council gives is that they free up the students' time for exploring and thinking about math concepts, not to add another layer of busywork to the school day.

"If teachers learned to teach by studying books and memorizing techniques"--the conscious, explicit way most grown-ups learn to use, say, their computers--"written recommendations might have their intended effect," write Stigler and Hiebert. But most people don't learn to teach the way they learn to compute. They learn how to teach long before they ever set foot in a college of education, the way we all learn how to behave at Thanksgiving dinner--mostly implicitly and by example.

The American way of teaching, they explain, is a cultural system--a pattern students and teachers alike picked up years ago, have long since taken for granted, and rarely if ever think about. As at the Thanksgiving dinner table, everybody has been learning the script since before they knew how to talk. Thanksgiving dinner at home means, among other things, that everyone congratulates the cook and no one expects to pay for it afterward. Math class means that students follow procedures outlined by the teacher and then practice applying them over and over. No one expects to puzzle out the procedures themselves, let alone prove their validity.

Like them or not, cultural systems are stable. They can take a lot of punishment without changing more than absolutely necessary. If you grew up and went through decades of school in which math consisted of receiving definitions and applying procedures based on them, then that's how you'll teach it. If someone comes along and says, "Use calculators so students have more time to think," you may start using them--but you aren't likely to reevaluate everything you ever knew about math to make room for the conceptual understanding or exploration calculators are supposed to make possible. Instead you will turn the calculators into another procedure to learn, another cog in a familiar system. And presto! One more potentially disruptive reform idea has been harmlessly assimilated.

Suppose for a moment that you were a maverick teacher who took the NCTM recommendation to use calculators to heart and went through all the work and soul-searching that it implies. In that case, you would have changed the classroom script so much that even your "good" students wouldn't know how to act. You would be the educational equivalent of a Thanksgiving guest repeatedly pressing a $20 bill on his host. That's why blaming teachers doesn't help. Even if they know better, as individuals they can't move very far, let alone escape the sticky web of cultural expectations.

So, Stigler and Hiebert ask, how on earth do you change a cultural system? Their answer--now going beyond the TIMSS videotapes to their own recommendations--is, slowly and respectfully. They propose that we take seriously the idea of teaching as a profession. But their idea of professionalism has more to do with responsibilities than with perks. They're thinking of learned professions like law and medicine. Every practitioner can contribute to the assembled body of agreed-upon skills, and every practitioner is expected to go on learning from that growing body of skills throughout his or her career. "A profession is created not by certificates and censures but by the existence of a substantive body of professional knowledge, as well as a mechanism for improving it."

No such mechanism for improving teaching exists in this country. Some individual teachers know a lot, but they practice in isolation (and in the invisible clutches of the culture). Education is notorious as a field in which a chasm yawns between research and practice. Well-established research findings are little known and rarely followed in the classroom.

Stigler and Hiebert think that something like Japanese teachers' "lesson study," in which colleagues literally plan the course of a new lesson in detail, could provide a grassroots way to improve student learning. "In lesson study, groups of teachers meet regularly over long periods of time (ranging from several months to a year) to work on the design, implementation, testing, and improvement of one or several 'research lessons.'" After extensive preparation and rehearsal, eventually one of the teachers teaches the lesson. Then the group goes over that experience and suggests improvements based on it. Teachers become researchers.

This isn't at all like lesson planning as we know it. The Japanese teachers work on one lesson at a time, in meticulous detail. In one case Stigler and Hiebert describe, largely based on research by Makoto Yoshida, the research lesson involved teaching simple subtraction from numbers larger than ten. Over the course of weeks the teacher group discussed:

"The problem with which the lesson would begin, including such details as the exact wording and numbers to be used.

"The materials students would be given to use in trying to solve the problem.

"The anticipated solutions, thoughts, and responses that students might develop as they struggled with the problem.

"The kinds of questions that could be asked to promote student thinking during the lesson, and the kinds of guidance that could be given to students who showed one or another type of misconception in their thinking.

"How to use the space on the chalkboard. (Japanese teachers believe that organizing the chalkboard is a key ingredient to organizing students' thinking and understanding.)

"How to apportion the fixed time of the lesson--about forty minutes--to different parts of the lesson.

"How to handle individual differences in level of mathematical preparation among the students.

"How to end the lesson--considered a key moment in which students' understanding can be advanced."

Obviously anything remotely like this, in any school, would take time and cost money. It works better where there's a nationwide curriculum, as in Japan, so that the teachers' published reports on their lesson studies would have wide applicability. Also obviously, lesson study wouldn't pay off right away in higher test scores, even on well-designed tests. But if Americans came to see teaching as a profession open to continuous improvement, change could begin right away.

As far as I can tell, the only place where this has begun in the Chicago area is among the five fourth-grade teachers in suburban Northbrook's small District 31. Superintendent Paul Kimmelman, who was following Stigler and Hiebert's work before The Teaching Gap was published, invited teachers to submit proposals for a pilot lesson-study project. The fourth-grade teachers' proposal won, and they've been spending an extra (paid) hour a week this school year working up a lesson on reducing fractions, which they'll teach March 8. The experience has brought them in closer touch with one another's methods, says Kimmelman. "They say it has already affected how they teach their whole-fractions unit."

"In our view," write Stigler and Hiebert, "lesson study is not the kind of process in which teachers must first develop a list of capabilities and then begin to design improved lessons. Lesson study is, in fact, the ideal context in which teachers develop deeper and broader capabilities....The system that we are proposing is one that we desperately need if we are to look back a hundred years from now and recognize a history of gradual progress in improving teaching."

On any given fall Sunday, when professional quarterbacks leave the football field during a game, they huddle with coaches over photographs of the plays the team just ran. They need to figure out what went right, what went wrong, and how to run the plays better next time. Until teachers are enabled to take every elementary math and English lesson that seriously, no school reform is going to reform much of anything.

The Teaching Gap: Best Ideas From the World's Teachers for Improving Education in the Classroom by James Stigler and James Hiebert, the Free Press, $23.

Art accompanying story in printed newspaper (not available in this archive): illustration/Slug Signorino.